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Flow solver: Proust / TurbFlow
The equations solved are the 3D, unsteady, compressible, Reynolds averaged (RANS) or spatially filtered (LES), Navier-Stokes equations cast in the absolute frame where the laminar viscosity is assumed constant or calculated by the Sutherland’s law.
1 Spatial discretization. The space discretization is based on a MUSCL finite volume formulation with moving structured meshes, which utilizes vertex variable storage. The convective flixes are evaluated using a 3rd order upwind scheme (Van Leer’s Flux Vector Splitting with the Hanel correction, Roe’s Approximate Riemann Solver, or Liou’s Advection Upwind Splitting Method), or 4th order centered scheme (for LES). An hybrid method combining the advantages of the central scheme in subsonic regions with the properties of the upwind scheme through discontinuities has been introduced to reduce the numerical losses in very low Mach number regions. The viscous terms are computed by a second order centered scheme. The resulting semidiscrete scheme is integrated in time using an explicit five steps Runge-Kutta time marching algorithm.
2 Boundary conditions. Compatibility relations are used to take into account physical boundary conditions. The outgoing characteristics are retained, since these provide information from inside the domain. The incoming characteristics, on the other hand, are replaced by physical boundary conditions, i. e. total pressure, total temperature and ft>w angles for a subsonic inlet, static pressure for a subsonic outlet, zero normal velocity component for a slip wall and zero velocity and heat fhx for an adiabatic wall. Ghost cells for which the equations are not solved are built around the domain to simulate geometrical boundary conditions, like periodicity and symmetry. Non refhctive boundary conditions are implemented by retaining the equations associated to the incoming characteristics, in which the wave velocity is fixed to zero to prohibit propagation directed into the computational domain.
3 Turbulence Models. For the RANS approach, turbulence is taken into account by a k — и model. Two transport equations are implemented, governing the turbulent energy k and the dissipation u. The evaluation of the Reynolds tensor and of the turbulent viscosity are carried out by different turbulence models: the linear model of Wilcox, 1993b, the low-Reynolds model
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Figure 1. VKI turbine. Left: Mach numbers contour’s. Right: Comparison of density isocontours [Upper : linear k-w model; Middle : experimental data (ONERA); Lower : non-linear k-w model] |
of Wilcox, 1993a, the non-linear model of Shih et al., 1995 and the non-linear model of Craft et al., 1996.
For the LES, the subgrid scales are represented by a viscosity, computed using the auto-adaptive model of Casalino et al., 2003. This formulation enables an effective evaluation of the subgrid-scale viscosity, even for complex geometries.
